The unseeded Slovakian took the first set 7-6 (7-5) before Kvitova bounced back to take the next 6-3. Sramkova, however, battled through injury to win the final set 6-4 and upset the two-time Wimbledon winner.
Seventh seed Nina Stojanovic avoided an early exit as she recovered from a first-set loss to beat Vitalia Diatchenko 6-7 (5-7) 6-1 6-3, while ninth seed Greet Minnen defeated Italian Giulia Gatto-Monticone 6-0 4-6 6-1.
Unlike Kvitova, the top seed at the Hungarian Grand Prix, Yulia Putintseva, cruised past Ekaterine Gorgodze 6-2 6-4 to record a straight-sets first-round win in Budapest.
But Irina-Camelia Begu, the fourth favourite, was not as fortunate as she failed to secure second-round progression, falling 3-6 4-6 to the unseeded Anna Karolina Schmiedlova.
The Hungarian Grand Prix provided more shocks, with Irina Bara knocking out eighth seed Sara Errani courtesy of a 4-6 7-5 6-2 triumph, while seventh seed Viktoriya Tomova crashed out to Paula Ormaechea in straight sets.
On day one of the Ladies Open Lausanne, favourite and 2021 Roland Garros semi-finalist Tamara Zidansek shook off a first-set loss to dispatch of Marina Melnikova 2-6 6-3 6-2 and book her place in the next round.
Meanwhile, second seed Fiona Ferro was leading her French compatriot Harmony Tan 7-5 2-1 before play was suspended for the day in Switzerland.